The rational design of prosthetic heart valves requires a theoretical technique for the prediction of the flow pattern of blood in the heart and for the analysis of the interaction of that flow pattern with the motions of the valves and heart muscle. This project involves the development of computational methods for the Navier-Stokes equations in three-dimensions, the computational description of the fiber architecture of the left ventricle and of the mitral and aortic valves based on the joint use of mechanics and differential geometry, and the development of a unified method which combines these techniques for the heart valve problem in three-dimensions. The project also involves the application of our current, two-dimensional methods to study the role of the following factors in heart valve performance: diastolic tension on the chordae tendineae, geometry of the mitral leaflets, contraction of the mitral annulus, timing and strength of atrial systole, exercise, and Reynolds number. Several of these studies will emphasize the comparison between the response of the prosthetic and natural mitral valves under identical physiological or pathophysiological interventions. We also plan a systemtic study of the fundamental parameters of ball and disc valves. For ball valves, these are the translation distance and the size of the ball; for disc valves they are the translation distance, the position of the pivot point, and the maximum angle of opening. Because the methods of this project predict not only the motions of the blood but also the deformations of the heart wall, they can be used to assess the influence of prosthetic valves on the cardiac muscle as well as on the hemodynamics. Because the methods predict the details of the flow pattern, they can be used to assess the influence of prosthetic valves on the cardiac endothelium and on the formed elements of the blood.